Exercise 2: Inhomogeneous Differential Equation with Initial Conditions
Maple exercise.
We are given the inhomogeneous differential equation
\begin{equation}
x^{\prime \prime}(t)+4x’(t)+29x(t)=-25\sin(2t)+\frac{109}{4}\mathrm e^{-\frac 12 t}-8\cos(2t), \quad t \in \reel.
\end{equation}
A
Find using Maple’s dsolve the general solution to the differential equation.
B
Plot the solution whose graph passes through the point $(0,1)\,$ and whose tangent has a slope of $-\frac 52\,$ at $\,t=0\,$. Then plot the solution whose graph also passes through the point $(0,1)\,,$ but whose tangent has a slope of $\,\frac 12\,$ at $\,t=0\,$.
is linearly independent (this has been proven in exercises from a previous week), consider the restriction of $\,f\,$ to the 5-dimensional subspace $\,U\,$ in $\,C^{\infty}(\reel)\,$ that has $\,v\,$ as a basis.
D
Show that the image $\,f(U)\,$ is a subspace in $\,U\,,$ and determine the mapping matrix $\,\matind vFv\,$ of the map $f:U\rightarrow U\,$ with respect to basis $\,v\,.$
hint
$\,f(U)\,$ is spanned by the images of the basis vectors.
Does a particular solution $\,x_0(t) \in U\,$ exist to the equation
$$\,f(x(t))=q(t)\,$$
that satisfies the initial conditions $\,x_0(0)=0\,$ and $\,x_0’(0)=1\,?$
answer
No.
Exercise 4: Modelling of a Physical Scenario
You will now model a physical scenario using Maple, and you will use the model for experimentation. The procedure is that you execute the Maple commands one at a time - so do not use the execution button !!! in Maple that executes the entire sheet at once. Fields with XX must be replaced with your own Maple command along the way. When you have finished an answer, then you are welcome to open the Solution tab for a suggested solution.
A
Download the file eMaple3 and have fun with the modelling.
Exercise 5: Uniqueness of the Solution. Theory
About a differential equation on the form
\begin{equation}
x^{\prime \prime}(t)+a_1x’(t)+a_0x(t)=q(t), \quad t \in \reel
\end{equation}
we are informed that $\,x_1(t)=\sin(t)\,$ and $\,x_2(t)=\frac{1}{2}\sin(2t)\,$ both are solutions.
A
Prove using the existence and uniqueness theorem that this statement is false.
hint
Use Theorem 18.20 in eNote 18.
hint
Do numbers $\,t_0\,$ that satisfy both $\,x_{1}(t_0)=x_{2}(t_0)\,$ and $\,x_{1}’(t_0)=x_{2}’(t_0)\,$ exist?
answer
When you have found a number $t_0$ that satisfies the two equations in the last hint, it follows from the
existence and uniqueness theorem that the statement is false.
Exercise 6: Structure of Solutions. Theory
We are given the inhomogeneous differential equation
\begin{equation}
x^{\prime \prime}(t)+a_1x’(t)+a_0x(t)=q(t), \quad t \in \reel
\end{equation}
together with two particular solutions,
\begin{equation}
x_1(t)=\sin t+2\e^t \quad \mathrm{and} \quad x_2(t)=\sin t+\e^t-\e^{-t}.
\end{equation}
A
Determine the general solution to the homogeneous equation.
hint
How would you utilize the information that the two functions are both a particular solution to the inhomogeneous equation?
hint
The difference between the two functions is a particular solution to the homogeneous equation.
hint
Find $x_1(t)-x_2(t)$. The roots of the characteristic equation are the exponents in this function. Moreover this function is a solution to the corresponding homogeneous equation.
Determine the general solution to the inhomogeneous equation.
C
Determine $\,a_0,\,a_1\,$ and $\,q(t)\,.$
Exercise 7: The Complex Guessing Method
A
Determine a particular solution to the complex differential equation
\begin{equation}
x^{\prime \prime}(t)-2x’(t)-3x(t)=10\,\e^{(-1+2i)t}, \quad t \in \reel.
\end{equation}
hint
Guess a function of the type $\,x_0(t)=c\e^{(-1+2i)t}\,.$
answer
$$x_0(t)=\left(-\frac 12 +i\right)\e^{(-1+2i)t}$$
B
Determine a particular solution to the differential equation
\begin{equation}
x^{\prime \prime}(t)-2x’(t)-3x(t)=10\,\e^{-t}\,\cos(2t), \quad t \in \reel.
\end{equation}
hint
Note that the right-hand side is the real part of the right-hand side in a).
hint
The real part of the solution in a) is a solution.
Determine a particular solution to the differential equation
\begin{equation}
x^{\prime \prime}(t)-2x’(t)-3x(t)=10\,\e^{-t}\,\sin(2t), \quad t \in \reel.
\end{equation}
hint
Note that the right-hand side is the imaginary part of the right-hand side in a).
Exercise 8: From the Solution to the Differential Equation
All real solutions to an inhomogeneous linear differential equation of second order are
\begin{equation}
x(t)=c_1\e^{-t}\cos 2t+c_2\e^{-t}\sin 2t+5t^3+t^2+12t+7,\quad (c_1,c_2)\in\reel^2.
\end{equation}
A
State the differential equation.
hint
When you have the general solution to the inhomogeneous equation, you can directly read the general solution to the corresponding homogeneous equation from that.
hint
The roots to the characteristic equation can be found within the general solution to the corresponding homogeneous equation.
hint
The homogeneous equation is
$$x^{\prime \prime}(t)+2x'(t)+5x(t)=0\,.$$
The left-hand side is also the left-hand side of the inhomogeneous equation. A particular solution to the inhomogeneous equation can be seen to be:
$$x_0(t)=5t^3+t^2+12t+7\,.$$
The right-hand side of the inhomogeneous equation can be found by substituting the particular solution into the left-hand side:
